194 research outputs found
Coloring d-Embeddable k-Uniform Hypergraphs
This paper extends the scenario of the Four Color Theorem in the following
way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly)
embedded into R^d. We investigate lower and upper bounds on the maximum (weak
and strong) chromatic number of hypergraphs in H(d,k). For example, we can
prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak
chromatic number is Omega(log n/log log n), whereas the weak chromatic number
for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.Comment: 18 page
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
Coloring face hypergraphs on surfaces
AbstractThe face hypergraph of a graph G embedded on a surface has the same vertex set as G and its edges are the sets of vertices forming faces of G. A hypergraph is k-choosable if for each assignment of lists of colors of sizes k to its vertices, there is a coloring of the vertices from these lists avoiding a monochromatic edge.We prove that the face hypergraph of the triangulation of a surface of Euler genus g is O(g3)-choosable. This bound matches a previously known lower bound of order Ω (g3). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O(gr)-choosable. Note that colorings of face hypergraphs for r=2 correspond to usual vertex colorings and the upper bound O(g) thus follows from Heawood’s formula. Separate results for small genera are presented: the bound 3 for triangulations of the surface of Euler genus g=3 and the bound 7+36g+496 for surfaces of Euler genus g≥3. Our results dominate the previously known bounds for all genera except for g=4,7,8,9,14
Approximating acyclicity parameters of sparse hypergraphs
The notions of hypertree width and generalized hypertree width were
introduced by Gottlob, Leone, and Scarcello in order to extend the concept of
hypergraph acyclicity. These notions were further generalized by Grohe and
Marx, who introduced the fractional hypertree width of a hypergraph. All these
width parameters on hypergraphs are useful for extending tractability of many
problems in database theory and artificial intelligence. In this paper, we
study the approximability of (generalized, fractional) hyper treewidth of
sparse hypergraphs where the criterion of sparsity reflects the sparsity of
their incidence graphs. Our first step is to prove that the (generalized,
fractional) hypertree width of a hypergraph H is constant-factor sandwiched by
the treewidth of its incidence graph, when the incidence graph belongs to some
apex-minor-free graph class. This determines the combinatorial borderline above
which the notion of (generalized, fractional) hypertree width becomes
essentially more general than treewidth, justifying that way its functionality
as a hypergraph acyclicity measure. While for more general sparse families of
hypergraphs treewidth of incidence graphs and all hypertree width parameters
may differ arbitrarily, there are sparse families where a constant factor
approximation algorithm is possible. In particular, we give a constant factor
approximation polynomial time algorithm for (generalized, fractional) hypertree
width on hypergraphs whose incidence graphs belong to some H-minor-free graph
class
Difference of Facial Achromatic Numbers between Two Triangular Embeddings of a Graph
A facial -complete -coloring of a triangulation on a surface is a vertex -coloring such that every triple of -colors appears on the boundary of some face of . The facial -achromatic number of is the maximum integer such that has a facial -complete -coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.
For two triangulations and on a surface, may not be equal to even if is isomorphic to as graphs. Hence, it would be interesting to see how large the difference between and can be. We shall show that an upper bound for such difference in terms of the genus of the surface
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Clique-Relaxed Graph Coloring
We define a generalization of the chromatic number of a graph G called the k-clique-relaxed chromatic number, denoted χ(k)(G). We prove bounds on χ(k)(G) for all graphs G, including corollaries for outerplanar and planar graphs. We also define the k-clique-relaxed game chromatic number, χg(k)(G), of a graph G. We prove χg(2)(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χg(2)(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χg(2)(H) ≤ 3
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